Official Blizzard Quote:
Spells, auras, abilities that buff, help, directly or indirectly heal yourself and party members, these are all good mechanics to helping cooperation in multiplayer games. And we'll certainly have some of them.
What we don't want though is a situation where you'd say "Ok just sit back and keep us healed." Everyone should be fighting, everyone should be DPS, and the buffs and bonuses that come from certain classes should simply be situational and/or beneficial "icing on the cake".
Blizzard has stated from the beginning that they are making Diablo III with cooperative play as its main stage. With cooperative play comes forming parties or groups. However, unlike a very famous Blizzard MMO, players will not have to worry about looking for a healer to form a party and complete certain obstacles. Blizzard wants every class to be able to destroy the demons of Hell whether they are in a group or the player is enjoying Single Player.
While no class is dedicated to healing or support, this is not to say that each class is not without some skills to help out themselves or their group stay alive as Bashiok mentioned. The Barbarian has skills like Strong Constitution and Death Proof to help keep him alive along with a slew shouts and battle cries to help out his group. The Witch Doctor has Jungle Fortitude to give him some extra HP and has Horrify and Mass Confusion to aid his allies. From the few skills revealed for the Monk, it is easy to see how he can help his group stay alive with Inner Sanctuary and Radiant Visage. Even the offensive heavy Wizard has Conjured Health to gain some need HP and Slow Time to help party members.
Along with no healing class, Blizzard has also removed the accessibility of potions for the player. They have created many more mechanics to help characters to constantly stay in the action of this game. Health Globes will help player heal as they continue to fight. The have also added checkpoints for when you do die and a co-op resurrection system. While some Hardcore Players may face a harder challenge without potions and no healing class, it is the challenge that keeps people playing Hardcore.
So for all of you who continue the fight against healers in Diablo III, you can rest. There will be no healing class in Diablo III.